A Liouville-type theorem for the coupled Schrödinger systems and the uniqueness of the sign-changing radial solutions

Abstract

In this paper, we study the sign-changing radial solutions of the following coupled Schrödinger system \begin{equation} \left\{ \begin{array}{lr} -Δu_j+λ_j u_j=μ_j u_j^3+\sum_{i\neq j}β_{ij} u_i^2 u_j \,\,\,\,\,\,\,\, \mbox{in }B_1 ,\nonumber u_j\in H_{0,r}^1(B_1)\mbox{ for }j=1,\cdots,N.\nonumber \end{array} \right. \end{equation} Here, ${\lambda }_j,{\mu }_j>0$ and ${\beta }_{ij}={\beta }_{ji}$ are constants for $i,j=1,\cdots ,N$ and $i\ne j$. $B_1$ denotes the unit ball in the Euclidean space ${\mathbb{R}}^3$ centred at the origin. For any $P_1,\cdots ,P_N\in \mathbb{N}$, we prove the uniqueness of the radial solution $(u_1,\cdots ,u_j)$ with $u_j$ changes its sign exactly $P_j$ times for any $j=1,\cdots ,N$ in the following case: ${\lambda }_j\ge 1$ and $|{\beta }_{ij}|$ are small for $i,j=1,\cdots ,N$ and $i\ne j$. New Liouville-type theorems and boundedness results are established for this purpose.

13 pages